Path degeneracy and applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912290455945216 |
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| author | Lin, Y. de Mendez, P. Ossona |
| author_facet | Lin, Y. de Mendez, P. Ossona |
| contents | In this work, we relate girth and path-degeneracy in classes with sub-exponential expansion, with explicit bounds for classes with polynomial expansion and proper minor-closed classes that are tight up to a constant factor (and tight up to second order terms if a classical conjecture on existence of $g$-cages is verified). As an application, we derive bounds on the generalized acyclic indices, on the generalized arboricities, and on the weak coloring numbers of high-girth graphs in such classes. Along the way, we prove a conjecture proposed in [T.~Bartnicki et al., Generalized arboricity of graphs with large girth, Discrete Mathematics 342 (2019), no.~5, 1343--1350.], which asserts that, for every integer $k$, there is an integer $g(p,k)$ such that every $K_k$ minor-free graph with girth at least $g(p,k)$ has $p$-arboricity at most $p+1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18614 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Path degeneracy and applications Lin, Y. de Mendez, P. Ossona Combinatorics Discrete Mathematics In this work, we relate girth and path-degeneracy in classes with sub-exponential expansion, with explicit bounds for classes with polynomial expansion and proper minor-closed classes that are tight up to a constant factor (and tight up to second order terms if a classical conjecture on existence of $g$-cages is verified). As an application, we derive bounds on the generalized acyclic indices, on the generalized arboricities, and on the weak coloring numbers of high-girth graphs in such classes. Along the way, we prove a conjecture proposed in [T.~Bartnicki et al., Generalized arboricity of graphs with large girth, Discrete Mathematics 342 (2019), no.~5, 1343--1350.], which asserts that, for every integer $k$, there is an integer $g(p,k)$ such that every $K_k$ minor-free graph with girth at least $g(p,k)$ has $p$-arboricity at most $p+1$. |
| title | Path degeneracy and applications |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2503.18614 |